Flow at high Reynolds numbers through anisotropic porous media

An approximate expression is developed for the relationship between the hydraulic gradient (J), the specific discharge (q) and fluid and porous matrix properties in the case of saturated, steady and uniform (macroscopic) flow of a Newtonian liquid at high Reynolds numbers through a homogeneous anisotropic porous medium:

$\text{g}\text{J}\text{=(v}\text{w}{}^{\mathrm{(2)}}\text{+}\text{B}{}^{\mathrm{(4)}}\text{:}\text{qq}\text{/q+}\text{C}{}^{\mathrm{(3)}}\text{·}\text{q}\text{)·}\text{q}$

In this expression, the tensors w⁽²⁾, B⁽⁴⁾ and C⁽³⁾ denote properties of the solid matrix only. The tensors W⁽²⁾, and C⁽³⁾ are symmetrical; the tensor B⁽⁴⁾ is symmetrical only in the first and last pairs of indices. It seems that no mathematical expression with a finite number of parameters exists, which can serve as a universal exact expression for the sought relationship between J and q.